Today is our topic of discussion Homogeneous, Symmetric and Cyclic Expressions .
Homogeneous, Symmetric and Cyclic Expressions
Homogeneous Polynomial:
If each term of a polynomial has the same degree, it is called homogeneous polynomial. The expression x² + 2xy + 5y² is a homogeneous polynomial of the variable x, y with two degree (each term having degree 2).
It is a special case of the homogeneous polynomial ax²+2hxy+by² of degree two in two variables, y where, a, h, b are definite numbers. Considering x, y, a, h, b the variables, ar² + 2hxy + by² is homogeneous polynomial of degree 3. 2x²y + y²z+9z2x-5xyz is a homogeneous polynomial of the variables x, y, z each term having degree 3.
(Symmetric Expression):
An expression with more than one variable, which remains unchanged when any two of its variables are interchanged is called symmetric expression.
The expression a+b+c is symmetric expression of the variables, a, b, c because the expression remains unchanged when any two of the variables are interchanged.Similarly, abbcca of the variables a, b, c and x² + y² + z² + xy + yz + zx are symmetric expression of the variables x, y, z.
But 2x²+5xy+6y² is not a symmetric expression of the variables x and y Because interchanging x and y it becomes 2y²+5xy + 6x² which is different from former expression.
(Cyclic Expression):
An expression with three variables, which remain unchanged when the first variable is replaced by the second, the second variable is replaced by the third and the third variable is replaced by the first variable is called (cyclically symmetric expression). If the replacing of the variables is done cycle-wise like the adjacent figure, it is called cyclically symmetric expression.
x² + y²+z² + xy+yz + zx is a cyclic expression of the variables x, y, z because replacing by y, y by z and z by r the expression remains the same.Similarly the expression x²y + y²z + z²x is a cyclic expression of variable x, y, z. The expression x² – y²+z2 is not cyclic because replacing x by y, y by z and z by a the expression becomes y² – z² + x² which is different from the former expression.
Clearly, every symmetric expression in three variables is cyclic. But not every cyclic expression is symmetric. For example, the expression, x²(y – z) + y²(z – x) + z²(x − y) is cyclic, not symmetric. For interchanging x and y it becomes – y²(x-2)+x²(z-y)+22 (y-x), which is different from the former expression.
Observation:
For the convenience of description, the expression of variable x, y is indexed by F(x, y) and that of variable x, y, z is indexed by F(x, y, z)
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